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Unformatted text preview: Chapter 3 Linear Algebra 3.1 Fields Consider a set F of objects and two operations on the elements of F , addition and multiplication. For every pair of elements s,t ∈ F then ( s + t ) ∈ F . For every pair of elements s,t ∈ F then st ∈ F . Suppose that these two operations satisfy 1. addition is commutative: s + t = t + s ∀ s,t ∈ F 2. addition is associative: r + ( s + t ) = ( r + s ) + t ∀ r,s,t ∈ F 3. to each s ∈ F there exists a unique element ( s ) ∈ F such that s +( s ) = 0 4. multiplication is commutative: st = ts ∀ s,t ∈ F 5. multiplication is associative: r ( st ) = ( rs ) t ∀ r,s,t ∈ F 6. there is a unique nonzero element 1 ∈ F such that s 1 = s ∀ s ∈ F 7. to each s ∈ F there exists a unique element s 1 ∈ F such that ss 1 = 1 8. multiplication distributes over addition: r ( s + t ) = rs + rt ∀ r,s,t ∈ F . Then, the set F together with these two operations is a feld . Example 3.1.1. The real numbers with the usual operations of addition and multi plication form a Feld. The complex numbers with these two operations also form a Feld. 34 3.2. MATRICES 35 Example 3.1.2. The set of integers with addition and multiplication is not a Feld. P 3.1.3. Is the set of rational numbers a subFeld of the real numbers? Example 3.1.4. Is the set of all real numbers of the form s + t √ 2 , where s and t are rational, a subFeld of the complex numbers? The set F = s + t √ 2 : s,t ∈ Q together with the standard addition and multiplication is a Feld. Let s,t,u,v ∈ Q , s + t √ 2 + u + v √ 2 = ( s + u ) + ( t + v ) √ 2 ∈ F s + t √ 2 u + v √ 2 = ( su + 2 tv ) + ( sv + tu ) √ 2 ∈ F s + t √ 2 1 = s t √ 2 s 2 + 2 t 2 = s s 2 + 2 t 2 t s 2 + 2 t 2 √ 2 ∈ F Again, the remaining properties are straightforward to prove. The Feld s + t √ 2 , where s and t are rational, is a subFeld of the complex numbers. 3.2 Matrices Let F be a feld and consider the problem oF fnding n scalars x 1 ,...,x n which satisFy the conditions a 11 x 1 + a 12 x 2 + ··· + a 1 n x n = y 1 a 21 x 1 + a 22 x 2 + ··· + a 2 n x n = y 2 . . . . . . . . . . . . a m 1 x 1 + a m 2 x 2 + ··· + a mn x n = y m (3.1) where { y i : 1 ≤ i ≤ n } ⊂ F and { a ij : 1 ≤ i ≤ m, 1 ≤ j ≤ n } ⊂ F . These conditions Form a system of m linear equations in n unknowns . A shorthand notation For (3.1) is the matrix equation Ax = y , where A is the matrix representation given by A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a mn 36 CHAPTER 3. LINEAR ALGEBRA and x , y denote x = ( x 1 ,...,x n ) T y = ( y 1 ,...,y m ) T . Defnition 3.2.1. Let A be an m × n matrix over F and let B be an n × p matrix over F . the matrix product AB is the m × p matrix C whose i,j entry is c ij = n r =1 a ir b rj ....
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 Fall '08
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 Linear Algebra, Vector Space, WI, Scalars

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